A033631 Numbers k such that sigma(phi(k)) = sigma(k) where sigma is the sum of divisors function A000203 and phi is the Euler totient function A000010.

1, 87, 362, 1257, 1798, 5002, 9374, 21982, 22436, 25978, 35306, 38372, 41559, 50398, 51706, 53098, 53314, 56679, 65307, 68037, 89067, 108946, 116619, 124677, 131882, 136551, 136762, 138975, 144014, 160629, 165554, 170037, 186231, 192394, 197806

Offset: 1

Author

Jud McCranie

Keywords

• nonn

Comments

For corresponding values of phi(k) and sigma(k), see A115619 and A115620.

This sequence is infinite because for each positive integer k, 3^k*7*1979 and 3^k*7*2699 are in the sequence (the proof is easy). A108510 gives primes p like 1979 and 2699 such that for each positive integer k, 3^k*7*p is in this sequence. - Farideh Firoozbakht, Jun 07 2005

There is another class of [conjecturally] infinite subsets connected to A005385 (safe primes). Examples: Let s,t be safe primes, s<>t, then 3^2*5*251*s, 2^2*61*71*s, 2*61*s*t and 2*19*311*s are in this sequence. So is 3*s*A108510(m). (There are some obvious exceptions for small s, t.) - Vim Wenders, Dec 27 2006

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 87, p. 29, Ellipses, Paris 2008.
• R. K. Guy, Unsolved Problems in Number Theory, B42.
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books 1997.
• David Wells, Curious and Interesting Numbers (Revised), Penguin Books, page 114.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• J.-M. De Koninck, F. Luca, Positive integers n such that sigma(phi(n))=sigma(n), JIS 11 (2008) 08.1.5.
• S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Crossrefs

Cf. A000203, A000010, A006872, A115619, A115620.

Programs

Magma

[k:k in [1..200000]| DivisorSigma(1,EulerPhi(k)) eq DivisorSigma(1,k)]; // Marius A. Burtea, Feb 09 2020

Mathematica

Do[If[DivisorSigma[1, EulerPhi[n]]==DivisorSigma[1, n], Print[n]], {n, 1, 10^5}]

PARI

is(n)=sigma(eulerphi(n))==sigma(n) \\ Charles R Greathouse IV, Feb 13 2013

Extensions

Entry revised by N. J. A. Sloane, Apr 10 2006